Journal of Constructional Steel Research 62 (2006) 893–905 www.elsevier.com/locate/jcsr Buckling strength of multi-story sway, non-sway and partially-sway frames with semi-rigid connections Georgios E. Mageirou ∗ , Charis J. Gantes 1 Laboratory of Metal Structures, Department of Structural Engineering, National Technical University of Athens, 9 Heroon Polytechniou, GR-15780, Zografou, Athens, Greece Received 2 August 2005; accepted 30 November 2005 Abstract The objective of this paper is to propose a simpliﬁed approach to the evaluation of the critical buckling load of multi-story frames with semi- rigid connections. To that effect, analytical expressions and corresponding graphs accounting for the boundary conditions of the column under investigation are proposed for the calculation of the effective buckling length coefﬁcient for different levels of frame sway ability. In addition, a complete set of rotational stiffness coefﬁcients is derived, which is then used for the replacement of members converging at the bottom and top ends of the column in question by equivalent springs. All possible rotational and translational boundary conditions at the far end of these members, featuring semi-rigid connection at their near end as well as the eventual presence of axial force, are considered. Examples of sway, non-sway and partially-sway frames with semi-rigid connections are presented, where the proposed approach is found to be in excellent agreement with the ﬁnite element results, while the application of codes such as Eurocode 3 and LRFD leads to signiﬁcant inaccuracies. c  2005 Elsevier Ltd. All rights reserved. Keywords: Buckling; Effective length; Stiffness coefﬁcients; Multi-story sway; Non-sway and partially-sway frames; Semi-rigid connections 1. Introduction Nowadays, the buckling strength of a member can be evaluated using engineering software based on linear or also non-linear (in terms of large displacements and/or material yielding) procedures with analytical or numerical methods [15]. Nonetheless, the large majority of structural engineers still prefer analytical techniques such as the effective length and notional load methods [26]. These two methodologies are included in most modern structural design codes (for example, Eurocode 3 [9], LRFD [23]). The objective of this work is to propose a simpliﬁed approach for the evaluation of critical buckling loads of multi- story frames with semi-rigid connections, for different levels of frame sway ability. To that effect, a model of a column in a multi-story frame is considered as individual. The contribution of members converging at the bottom and top ends of the ∗ Corresponding author. Tel.: +30 210 9707444; fax: +30 210 9707444. E-mail addresses: mageirou@central.ntua.gr (G.E. Mageirou), chgantes@central.ntua.gr (C.J. Gantes). 1 Tel.: +30 210 7723440; fax: +30 210 7723442. column is taken into account by equivalent springs. Namely, the restriction provided by the other members of the frame to the rotations of the bottom and top nodes is modeled via rotational springs with constants c b and c t , respectively, while the resistance provided by the bracing system to the relative transverse translation of the end nodes is modeled via a translational spring with constant c br . This is shown schematically in Fig. 1. The rotational stiffness of the springs must be evaluated considering the inﬂuence of the connection non-linearity. This model has been used by several investigators (for example, Wood [27], Aristizabal-Ochoa [1], and Cheong- Siat-Moy [6]) for the evaluation of the critical buckling load of the member, and is adopted by most codes. The stiffness of the bottom and top rotational springs is estimated by summing up the contributions of members converging at the bottom and top ends, respectively: c b =  i c b,i , c t =  j c t , j . (1) A frame is characterized as non-sway if the stiffness c br of the bracing system is very large, as sway if this stiffness is negligible, and as partially-sway for intermediate values of this 0143-974X/$ - see front matter c  2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2005.11.019